Questions tagged [quivers]

"Quiver" is the word used for "directed graph" in some parts of representation theory. The main reason to use the term quiver is to indicate an interest in considering representations of the quiver.

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Projective resolution of a quiver with relations

How do we compute the projective resolution of a representation of a quiver with relations. For example consider the Beilinson quiver $B_4$ $. with the relations ­$\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
1 vote
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123 views

A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below) I want to ...
3 votes
0 answers
89 views

Auslander-Reiten sequences where irreducible morphisms are all epi/mono

Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles. Let $M$ be such a ...
18 votes
1 answer
541 views

Is there a cotangent bundle of a stable $\infty$-category?

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following? When $C$ is the ...
4 votes
0 answers
146 views

Proof of McKay correspondence without classifications

$\DeclareMathOperator\SU{SU}$I am wondering if there is any known proof of the McKay correspondence (I will give the precise statement that I mean by this) that doesn't use the classification of ...
9 votes
1 answer
217 views

Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
0 votes
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62 views

geometric objects in quiver variety corresponding to short exact sequences

I was studying quiver variety and known that representations of a quiver correspond to points in the corresponding quiver variety. So if give you a fixed triple representations $(M_1,M_2,M_3)$, I was ...
0 votes
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67 views

"Approximating" ring of semi-invariants

I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
1 vote
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230 views

Confusion regarding the invariant rational functions

I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below) It says that "every invariant rational function can ...
6 votes
0 answers
719 views

Tensor product of quivers

As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Probably this ...
1 vote
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126 views

Number of cluster variables associated to A type quivers

In a seminar/reading course about cluster algebras, we came across the fact that the number of cluster variables for the cluster algebra associated to mutating the quiver $A_n$ is $n(n+3)/2$ (rather, ...
10 votes
2 answers
2k views

Derived category of varieties and derived category of quiver algebras

I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ ...
5 votes
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98 views

Modern proof of a theorem of Dickson on finite representation type

In Theorem 3.1 the paper S. Dickson, On algebras of finite representation type Trans. Amer. Math. Soc. 135 (1969), 127-141, Dickson gives a sufficient condition for an algebra to have infinite ...
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86 views

Connected components of $Q(\mathrm{s\tau-tilt}A)$

I'm reading about support $\tau$-tilting modules and their mutations. I'm trying to understand the mutation quiver. Let $A$ be a finite dimensional algebra over an algebraically closed field, which is ...
2 votes
1 answer
213 views

Tame/wild classification of *cyclic* quivers?

There is a famous classification of the path algebras of finite acyclic quivers into finite, tame, and wild representation types. For quivers with cycles, it is standard that the 2-loop quiver (with ...
2 votes
0 answers
83 views

Example of a triangular string algebra that is rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. Hence, $A$ can be realized as the path algebra of a bound quiver $(Q,I)$, where $I\subseteq\mathbb{K}Q$ is an ...
2 votes
1 answer
140 views

Rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting ...
1 vote
0 answers
59 views

When the dg cluster category of a quiver is saturated?

Let $Q$ be a finite quiver without oriented cycles. In https://arxiv.org/abs/0807.1960 , Keller defines the dg cluster category $C_Q$ of $Q$. When is $C_Q$ smooth proper dg-category? If $Q$ is a ...
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50 views

Rank of Coxeter matrix

Let $Q$ be a quiver and $\Phi_Q$ be the Coxeter matrix of $Q$. Then $\Phi_Q\pm I$ are full-rank?
1 vote
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115 views

Quiver representations and the standard matrix decompositions

Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form: $$M = A D B$$ where $D$...
6 votes
2 answers
890 views

Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?

A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
1 vote
0 answers
58 views

Structure of tame concealed algebra of Euclidean type

I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...
5 votes
1 answer
507 views

Morita equivalence of acyclic categories

(Crossposted from math.SE.) Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. ...
2 votes
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57 views

Number of admissible quotient algebras

Let $Q$ be a finite connected quiver. An admissible quotient algebra is an algebra of the form $KQ/I$ with an admissible ideal $I$. Question 1: Is there a nice closed formula for the number of ...
1 vote
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78 views

When is $Y$ not an orbit closure?

Let $G$ be a linearly reductive algebraic group acting regularly on an affine space over $\mathbb{A}^n$ an algebraically closed field $\mathbb{K}$. Let $Y$ be a $G$-invariant (closed) affine ...
3 votes
1 answer
102 views

References on coefficient quivers

I would like to study about coefficient quivers, but I cannot find a good reference, as book for example. I could find many papers working with coefficient quivers, but none of them give a book or a &...
2 votes
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118 views

Two notions of stability

Let $Q$ be a finite quiver (i.e. an oriented graph). A representation of $Q$ is by definition a module over the path algebra of $Q$. More concretely, a representation associates to every vertex $v \in ...
34 votes
8 answers
7k views

Are quivers useful outside of Representation Theory?

There is a trend, for some people, to study representations of quivers. The setting of the problem is undoubtedly natural, but representations of quivers are present in the literature for already >...
1 vote
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206 views

Is $\langle\chi,\lambda\rangle=0$, whenever the limit exists? Where is the mistake?

Suppose $G$ is a linearly reductive algebraic group acting linearly on a finite dimensional vector space $V$ over $\mathbb{C}$. This induces an action on the coordinate ring $\mathbb{C}[V]$ (see here)....
7 votes
3 answers
409 views

Smallest faithful representation of an upper-triangular matrix quotient

This is a curiosity question that came out of teaching abstract algebra. Let $F$ be a field, and $n>1$ an integer. Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices ...
2 votes
0 answers
118 views

Getting an equivariant morphism

Let $X\subset\mathbb{A}^n$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristic zero. Suppose we have two linearly reductive algebraic groups $G$, $G'$ ...
1 vote
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52 views

Is the Schofield semi invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
2 votes
1 answer
172 views

Orbits in the open set given by Rosenlicht's result

Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we ...
1 vote
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122 views

How to determine if an invariant rational function is defined at the $\theta$-polystable point

Background: Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{C}$. Assume there is a quiver $Q=(Q_0,Q_1)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of ...
5 votes
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347 views

What representation theoretic properties does the semi-invariant ring tell us?

I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)? I have been studying about semi-...
2 votes
0 answers
60 views

Quiver and relations for Hopf algebras associated to quiver algebras

Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$. $A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via $[x,y]=xy-yx$ and $x^{p}=x^p$. ...
4 votes
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232 views

Road map for learning cluster algebras

I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
3 votes
0 answers
101 views

Example of an irreducible component with an open set of infinitely many codimension 2 (codimension 3) orbits

Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, ...
2 votes
0 answers
98 views

When do two path algebras share an underlying graph?

Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction. Since ...
1 vote
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95 views

Is $U\subseteq X^{s}(L)$?

Let $X$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristics $0$. Let $G$ be a connected, linearly reductive, affine algebraic group acting regularly on $...
1 vote
0 answers
123 views

Example of a brick-infinite, tame, triangular algebra of global dimension$\geq 3$

I'm trying to compute some examples and I'm unable to come up with a following example: What is(are) the example(s) of an acyclic quiver $Q$ with relations such that the 2-Kronecker quiver is NOT a ...
5 votes
3 answers
777 views

Quiver with two objects and two arrows composing to zero

In the description of the integral Adams spectral sequence, representations of the following quiver (with relations) arise naturally: We have two objects $A, B$, we have two arrows $\pi: A \...
1 vote
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110 views

Prove that $B$ is a directing module

Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
13 votes
0 answers
610 views

The derived category of integral representations of a Dynkin quiver

Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
2 votes
0 answers
82 views

Quiver and relations for group algebras of p-groups

Let $G$ be a finite $p$-group and $K$ a field of characteristic $p$. $KG$ is isomorphic to a quiver algebra $KQ/I$ with admissible ideal $I$. Question 1: Does there always exist such an $I$ where the ...
2 votes
0 answers
242 views

Understanding a proof of a result of Schofield

I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's ...
0 votes
1 answer
231 views

Quiver representations over any commutative ring

I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image. Towards the end, he has this ...
1 vote
1 answer
207 views

A result of Schofield in the case of quivers with relations

Let $Q$ be a quiver without oriented cycles. A result of Schofield says that, for dimension vectors $\alpha$ and $\beta$ of $Q$, $\beta\hookrightarrow\alpha$ iff $\operatorname{ext}(\beta, \alpha-\...
4 votes
1 answer
99 views

How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?

The potential of the quiver associated to surfaces is the canonical one given by Labardini-Fragoso's 2009 paper, who proved that the the QP associated to surfaces whose boundary is nonempty is rigid ...
2 votes
2 answers
136 views

Finding exceptional regular representations of $\tilde{D}_4$ efficiently

Let $A$ be the path algebra of the quiver $\tilde{D}_4$. I would like to find its exceptional regular representations with as little computation as possible. Of course, we can compute the whole ...